Subharmonic Configurations and Algebraic Cauchy Transforms of Probability Measures

Details Subharmonic Configurations and Algebraic Cauchy Transforms of Probability Measures Subharmonic Configurations and Algebraic Cauchy Transforms of Probability Measures

2009-12-23

arxiv.org/abs/0912.4650v1

Mathematics

Complex Variables

20 pages, one figure

Scientific paper

We study subharmonic functions whose Laplacian is supported on a null set and in connected components of of the complement to the support admit harmonic extensions to larger sets. We prove that if such a function has a piecewise holomorphic derivative, then it is locally piecewise harmonic. In generic cases it coincides locally with the maximum of finitely many harmonic functions. Moreover, we describe the support when the holomorphic derivative satisfies a global algebraic equation. The proofs follow classical patterns and our methods may also be of independent interest.

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